Renormalization-group (RG) theory permits critical exponents to vary continuously along a line of critical points when the fixed point admits a marginal operator, but essentially all known realizations obey Suzuki's weak-universality pattern: exponents such as β\beta, γ\gamma, ν\nu vary while δ\delta and η\eta stay fixed. Khan, Sarkar, Midya, Mandal and Mohanty (Sci. Rep. 7, 45004 (2017)) measured the ferromagnetic transition of (Sm1−yNdy)0.52Sr0.48MnO3(\mathrm{Sm}_{1-y}\mathrm{Nd}_y)_{0.52}\mathrm{Sr}_{0.48}\mathrm{MnO}_3 single crystals and found that all three exponents β\beta, γ\gamma, δ\delta vary continuously with the Nd concentration yy — a violation of both universality and weak universality — which they described by a scaling ansatz in which a marginal parameter λ\lambda rescales the exponents of the parent (three-dimensional Heisenberg) universality class. They concluded by posing an open problem: the microscopic identification of the marginal interaction responsible for this beyond-weak-universality variation remains elusive; in particular, it is not understood how or why a marginal operator is generated under the experimental conditions studied — chemical substitution in manganites being the paradigm case — and the development of a microscopic theory accommodating the phenomenon is an outstanding challenge. As of 2026 the problem remains open: partial progress — a long-range-strain RG theory (Singh and Puri), a scaling-level superuniversality hypothesis (Mukherjee and Mohanty), and a quenched-disorder mechanism in a nonequilibrium toy model (Bhoyar, Rangarajan and Gade) — stops short of a microscopic identification.

Background

Near a continuous phase transition, thermodynamic quantities follow power laws in the reduced temperature ε=T/TC−1\varepsilon = T/T_C - 1 and the magnetic field HH: the spontaneous magnetization vanishes as MS∼(−ε)βM_S \sim (-\varepsilon)^{\beta}, the initial susceptibility diverges as χ∼∣ε∣−γ\chi \sim |\varepsilon|^{-\gamma}, and along the critical isotherm M(H,TC)∼H1/δM(H, T_C) \sim H^{1/\delta}. The correlation length ξ∼∣ε∣−ν\xi \sim |\varepsilon|^{-\nu} and the specific heat C∼∣ε∣−αC \sim |\varepsilon|^{-\alpha} complete the standard set of critical exponents, tied together by scaling relations such as the Widom relation δ=1+γ/β\delta = 1 + \gamma/\beta and hyperscaling 2−α=dν2 - \alpha = d\nu in dd dimensions. Universality — the statement that exponents and scaling functions depend only on symmetries and dimension, not on microscopic details — has one systematic loophole within renormalization-group (RG) theory: if the critical fixed point admits a marginal operator (an interaction whose RG eigenvalue is exactly zero, so its strength is neither amplified nor suppressed under coarse-graining), a line of critical points can exist along which the exponents vary continuously with the strength of that interaction. The classic exactly solved example is the two-dimensional eight-vertex model, equivalent to two coupled Ising layers with a marginal four-spin interaction (the Ashkin–Teller coupling). Suzuki's weak-universality hypothesis observes that in nearly all known cases the variation respects a restricted pattern: exponents such as β\beta, γ\gamma, ν\nu vary while their ratios — equivalently δ=1+γ/β\delta = 1 + \gamma/\beta and η\eta — remain fixed; this holds for the eight-vertex and Ashkin–Teller families, frustrated spin systems, interacting dimers, Blume–Capel models (spin-1 magnets with a tunable single-ion anisotropy), reaction–diffusion systems, absorbing-state transitions, and percolation-type models. Exceptions in which all exponents vary are rare; the best understood mechanism is Fisher's long-range-interaction route to nonuniversal exponents in micellar solutions, where interactions decaying as r−(d+σ)r^{-(d+\sigma)} with a tunable decay exponent σ\sigma yield exponents that depend continuously on σ\sigma (marginal at σ=0\sigma = 0).

Doped rare-earth manganites RE1−xAExMnO3\mathrm{RE}_{1-x}\mathrm{AE}_x\mathrm{MnO}_3 (RE a rare-earth ion, AE an alkaline-earth ion) are mixed-valence ferromagnets whose metallic ferromagnetism arises from double exchange — hopping of ege_g electrons (occupying the 3d3d ege_g orbital doublet of Mn) between Mn3+\mathrm{Mn}^{3+} and Mn4+\mathrm{Mn}^{4+} ions favored by parallel core spins — modified by Jahn–Teller lattice distortions (local oxygen-octahedron distortions coupled to the degenerate ege_g levels of Mn3+\mathrm{Mn}^{3+}), cation-size disorder on the perovskite A-site (the rare-earth/alkaline-earth sublattice), and long-range elastic strain. Their paramagnetic-to-ferromagnetic (PM–FM) transitions are notoriously sensitive to bandwidth and quenched disorder: they range from first order to continuous with exponents belonging to no standard universality class. Continuous evolution of exponents under chemical substitution had been reported in URu2−xRexSi2\mathrm{URu}_{2-x}\mathrm{Re}_x\mathrm{Si}_2 and in Sr1−zCazRuO3\mathrm{Sr}_{1-z}\mathrm{Ca}_z\mathrm{RuO}_3, but in a weak-universality-compatible or otherwise special way.

Khan et al. measured M(H,T)M(H,T) in (Sm1−yNdy)0.52Sr0.48MnO3(\mathrm{Sm}_{1-y}\mathrm{Nd}_y)_{0.52}\mathrm{Sr}_{0.48}\mathrm{MnO}_3 single crystals for y=0.5,0.6,0.8,1.0y = 0.5, 0.6, 0.8, 1.0. At y=1y = 1 the exponents (β,γ,δ)=(0.36,1.38,4.72)(\beta, \gamma, \delta) = (0.36, 1.38, 4.72) match the three-dimensional Heisenberg class (0.365, 1.386, 4.82); as yy decreases to 0.5 they move continuously to (0.16, 1.27, 9.30) — β\beta roughly halves, δ\delta roughly doubles, γ\gamma changes only slightly — while the Widom relation and single-equation-of-state scaling collapse hold at every yy, and the transition becomes discontinuous below y≈0.4y \approx 0.4. To describe this, they proposed a scaling ansatz: if (α0,ν0,γ0,δ0)(\alpha_0, \nu_0, \gamma_0, \delta_0) are the exponents of a parent universality class, a marginal parameter λ\lambda (with two exponents κ,ω\kappa, \omega) generates the family

ν=ν0λκ+ω,2−α=2−α0λκ+ω,γ=γ0λω,δ+1δ−1=1λκδ0+1δ0−1,\nu = \frac{\nu_0}{\lambda^{\kappa+\omega}},\quad 2-\alpha = \frac{2-\alpha_0}{\lambda^{\kappa+\omega}},\quad \gamma = \frac{\gamma_0}{\lambda^{\omega}},\quad \frac{\delta+1}{\delta-1} = \frac{1}{\lambda^{\kappa}}\frac{\delta_0+1}{\delta_0-1},

which preserves all scaling relations for any λ\lambda. The case κ=0\kappa = 0 (with δ\delta, η\eta fixed) reproduces weak universality; ω=0\omega = 0 reproduces a variation with γ\gamma fixed seen in strong-coupling QED; the generic case κ>0\kappa > 0 fits the manganite data with κ=1.29\kappa = 1.29 and a fitted λ(y)\lambda(y), and predicts a multicritical point at λ=γ0\lambda = \gamma_0 where β→0\beta \to 0 and the transition turns discontinuous — consistent with the observed first-order regime at low yy. The ansatz is, however, purely phenomenological: λ\lambda has no microscopic origin, and the correspondence between yy and λ\lambda is fitted, not derived. The open question posed in the paper's conclusions is exactly this gap: which microscopic interaction is the marginal one, how does chemical substitution generate it, and can a microscopic theory be built that accommodates the phenomenon.

Problem Statement

Identify, at the microscopic level, the marginal interaction responsible for the continuous variation of critical exponents beyond weak universality, and construct a microscopic theory that accommodates it. The paradigm experimental system is the substitution-tuned ferromagnetic series (Sm1−yNdy)0.52Sr0.48MnO3(\mathrm{Sm}_{1-y}\mathrm{Nd}_y)_{0.52}\mathrm{Sr}_{0.48}\mathrm{MnO}_3 (0.5≤y≤10.5 \le y \le 1), in which all three exponents β\beta, γ\gamma, δ\delta vary continuously with yy — from the three-dimensional Heisenberg values at y=1y = 1 to (β,γ,δ)≈(0.16,1.27,9.30)(\beta, \gamma, \delta) \approx (0.16, 1.27, 9.30) at y=0.5y = 0.5 — while the Widom relation and single-equation-of-state scaling hold at each yy, and the transition turns first order below y≈0.4y \approx 0.4; analogous substitution-driven variation is seen in other doped manganites and related compounds — the source paper poses the question for substitution-tuned magnets generally, including the URu2−xRexSi2\mathrm{URu}_{2-x}\mathrm{Re}_x\mathrm{Si}_2 and Sr1−zCazRuO3\mathrm{Sr}_{1-z}\mathrm{Ca}_z\mathrm{RuO}_3 series, but the manganite series is its central system and serves here as the concrete reference data set. A complete answer of the affirmative kind must deliver all four parts jointly, as one coherent objective: (i) start from the microscopic physics of these materials (double exchange, Jahn–Teller/electron-phonon coupling, A-site cation disorder, elastic strain) and exhibit an interaction that is marginal, or effectively marginal, under renormalization-group flow (marginal means an RG eigenvalue of exactly zero; effectively marginal means an RG eigenvalue that vanishes at a limiting value of a tunable microscopic parameter, as for long-range interactions that become marginal as their decay exponent approaches σ=0\sigma = 0); (ii) explain how and why chemical substitution generates and tunes this operator; (iii) derive, rather than fit, the mapping between the substitution parameter and the marginal parameter λ\lambda of the scaling ansatz, so that the observed pattern of exponent variation — β\beta nearly halving, δ\delta nearly doubling, γ\gamma nearly unchanged between y=1y = 1 and y=0.5y = 0.5, with the approach to a multicritical point (where a line of continuous transitions meets a first-order line) and the onset of discontinuous behavior — follows from the theory; and (iv) make testable predictions for related substitution series. An answer of the alternative kind — a quantitative demonstration that the observed beyond-weak-universality variation arises without any marginal operator (for example from crossover, inhomogeneity, Griffiths-type rare-region effects (slowly relaxing magnetically ordered regions inside the nominally disordered phase), or experimental-analysis artifacts) and reproduces the same data — would equally resolve the question as posed by the source.

References

  1. N. Khan, P. Sarkar, A. Midya, P. Mandal, and P. K. Mohanty, "Continuously Varying Critical Exponents Beyond Weak Universality", Scientific Reports 7, 45004 (2017). DOI: 10.1038/srep45004, https://doi.org/10.1038/srep45004 (preprint: arXiv:1604.07688, https://arxiv.org/abs/1604.07688)
  2. I. Mukherjee and P. K. Mohanty, "Hidden superuniversality in systems with continuous variation of critical exponents", Physical Review B 108, 174417 (2023). DOI: 10.1103/PhysRevB.108.174417, https://doi.org/10.1103/PhysRevB.108.174417 (preprint: arXiv:2307.15022, https://arxiv.org/abs/2307.15022)
  3. R. Singh and S. Puri, "Strain fields and critical phenomena in manganites I: spin-lattice Hamiltonians", Journal of Statistical Mechanics: Theory and Experiment 2023, 033205 (2023). DOI: 10.1088/1742-5468/acbe97, https://doi.org/10.1088/1742-5468/acbe97 (preprint: arXiv:2211.09384, https://arxiv.org/abs/2211.09384)
  4. R. Singh and S. Puri, "Strain fields and critical phenomena in manganites II: spin-lattice-energy Hamiltonians", Journal of Statistical Mechanics: Theory and Experiment 2023, 033206 (2023). DOI: 10.1088/1742-5468/acc065, https://doi.org/10.1088/1742-5468/acc065 (preprint: arXiv:2211.09387, https://arxiv.org/abs/2211.09387)
  5. P. D. Bhoyar, G. Rangarajan, and P. M. Gade, "Emergence of continuously varying critical exponents in coupled map lattice as an effect of quenched disorder", Chaos, Solitons & Fractals 201, 117253 (2025). DOI: 10.1016/j.chaos.2025.117253, https://doi.org/10.1016/j.chaos.2025.117253 (preprint: arXiv:2509.07529, https://arxiv.org/abs/2509.07529)
  6. M. R. Laouyenne, M. Baazaoui, Sa. Mahjoub, W. Cheikhrouhou-Koubaa, Kh. Farah, and M. Oumezzine, "Continuously varying of critical exponents with the bismuth doped in the La0.8Na0.2Mn1−xBixO3 (0 ≤ x ≤ 0.06) manganites", Journal of Magnetism and Magnetic Materials 451, 629–638 (2018). DOI: 10.1016/j.jmmm.2017.11.123, https://doi.org/10.1016/j.jmmm.2017.11.123
  7. M. E. Fisher, "Long-Range Crossover and 'Nonuniversal' Exponents in Micellar Solutions", Physical Review Letters 57, 1911–1914 (1986). DOI: 10.1103/PhysRevLett.57.1911, https://doi.org/10.1103/PhysRevLett.57.1911
  8. M. Suzuki, "New Universality of Critical Exponents", Progress of Theoretical Physics 51, 1992–1993 (1974). DOI: 10.1143/ptp.51.1992, https://doi.org/10.1143/ptp.51.1992
  9. P. K. Mukherjee, P. Sarkar, and A. K. Chattopadhyay, "Role of external and internal perturbations on ferromagnetic phase transitions in manganites: existence of tricritical points", Journal of Physics: Condensed Matter 30, 295101 (2018). DOI: 10.1088/1361-648X/aacb3a, https://doi.org/10.1088/1361-648X/aacb3a
  10. P. Sarkar, N. Khan, K. Pradhan, and P. Mandal, "Collapse of ferromagnetism with Ti doping in Sm0.55Sr0.45MnO3: a combined experimental and theoretical study", Physical Review B 98, 014422 (2018). DOI: 10.1103/PhysRevB.98.014422, https://doi.org/10.1103/PhysRevB.98.014422 (preprint: arXiv:1803.10588, https://arxiv.org/abs/1803.10588)
  11. D. Rossini and E. Vicari, "Stacked quantum Ising systems and quantum Ashkin-Teller model", Physical Review B 113, 155143 (2026). DOI: 10.1103/y9n7-d6vr, https://doi.org/10.1103/y9n7-d6vr (preprint: arXiv:2601.18922, https://arxiv.org/abs/2601.18922)
  12. P. Adelhardt and K. P. Schmidt, "Continuously varying critical exponents in long-range quantum spin ladders", SciPost Physics 15, 087 (2023). DOI: 10.21468/SciPostPhys.15.3.087, https://doi.org/10.21468/SciPostPhys.15.3.087 (preprint: arXiv:2209.01182, https://arxiv.org/abs/2209.01182)